Noncommutative algebraic dynamics: Ergodic theory for profinite groups

Infinite ergodic theory for Kleinian groups

In this paper we use infinite ergodic theory to study limit sets of essentially free Kleinian groups which may have parabolic elements of arbitrary rank. By adapting a method of Adler, we construct a section map S for the geodesic flow on the associated hyperbolic manifold. We then show that this map has the Markov property and that it is conservative and ergodic with respect to the invariant m...

Ergodic Theory, Groups, and Geometry

Profinite Groups

γ = c0 + c1p+ c2p + · · · = (. . . c3c2c1c0)p, with ci ∈ Z, 0 ≤ ci ≤ p− 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pZ for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if such an equation has a solution in the integers, then it...

Fusion systems for profinite groups

We introduce the notion of a pro-fusion system on a pro-p group, which generalizes the notion of a fusion system on a finite p-group. We also prove a version of Alperin’s Fusion Theorem for pro-fusion systems.

Noncommutative localisation in algebraic K-theory II

In [Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic K-theory I, Geom. Topol. 8 (2004) 1385–1425] we proved a localisation theorem in the algebraic K-theory of noncommutative rings. The main purpose of the current article is to express the general theorem of the previous paper in a more user-friendly fashion, in a way more suitable for applications. In the process we compa...